The aim of this paper is to present a teaching proposal for the theoretical part relating to the first- and second-order linear difference equations with constant coefficients suitable for the first-year students at various types of universities. In contradistinction to the methods often applied (memorization of algorithms without a proper…

A math walk is a planned walk during which students talk about mathematical concepts embedded in their everyday environments. The walk is based on relatively permanent objects or locations, such as trees, artworks, playground equipment, and building elements, and can be shared with the broader school community in a variety of different ways. In…

Asking students questions is a ubiquitous but often underestimated aspect of teaching within higher education. Questions can be used to engage students in discussion, to check their understanding, and to encourage participation or sharing of ideas. However, while it is a useful strategy to prepare a few key questions before class, many questions…

Inherent in the Common Core's Standard for Mathematical Practice to "look for and express regularity in repeated reasoning" (SMP 8) is the idea that students engage in this practice by generalizing (NGA Center and CCSSO 2010). In mathematics, generalizing involves "lifting" and communicating about ideas at a level where the…

In this article, the authors define learning goals, differentiate them from performance goals, and elaborate on their function and importance as the guiding force behind maintaining cognitive rigor during mathematics learning. Next, the authors present a framework that teachers can use to define and refine learning goals during lesson planning…

Mathematical paradoxes often produce awe and wonder in the mathematics classroom. In this classroom episode, I share a paradoxical task, based on Simpson's Paradox, and its power as an intervention for a child diagnosed with ADHD. The Paradox leveraged his strengths to help him build understandings in proportional reasoning.

Thinking about and inventing concepts is the hallmark of the philosophical endeavor. Concepts like fairness, beauty, friendship and knowledge are some of those with a long philosophical lineage that start with questions like 'What is …?', 'Why?', 'How is it different from …?' and 'Does it then follow that…?' These questions are crucial, as they…

As part of the "Next Generation Science Standards" (K-LS1-1), students are expected to develop an understanding of what living things need in order to live and grow. The lessons provided in this article help students understand the difference between living and nonliving things, and help them acquire important background knowledge to…

In this article, the author asserts that asking and responding to diagnostic questions is the single most important part of teaching secondary school mathematics. He notes the importance of formative assessment and recommends a formative assessment strategy that requires students to be public about their answers to questions, displaying their…

The vision of mathematics learning advocated by the National Council of Teachers of Mathematics (NCTM) for twenty-five years (1989, 1991, 2000, 2014) positions students as active learners, constructing their knowledge of mathematics through exploration, discussion, and reflection. World-class standards put into place by states and provinces over…

Paper folding is an easy and inexpensive way to engage students artistically, culturally, and mathematically. The pinwheel base is the source for many origami playthings, from a pinwheel to a boat to a butterfly. This article explores a fourth grade activity that repurposes the pinwheel base into a rich and inexpensive manipulative for the review…

This chapter describes a multidisciplinary faculty self-study about reciprocity in service-learning. The study began with each coauthor participating in a Decoding interview. We describe how Decoding combined with collaborative self-study had a positive impact on our teaching practice.

The purpose of this paper is to account for the advancement of questioning in mathematics education. Four sections focus on: (1) Importance of Questioning in Education (a historical perspective of questioning from the 1960s through the 1990s); (2) Cognitive Classification of Questions (Costa's Levels of Thinking, Webb's Depth of Knowledge, and…

Pupils' questions can play a powerful role in helping them to develop conceptual understanding, but can also provide a stimulus for teacher learning. This article presents a series of stories that illustrate the challenges of capitalising on pupils' questions, given the complexity of decision-making required in the science classroom and the…

According to Dylan Wiliam, the traditional classroom practice in which a teacher asks a question, students raise their hands, and the teacher calls on a volunteer does not actually provide much useful information--and it may even impede learning. When teachers ask questions in this way, they're only engaging the most confident students in the…